What is a zero, and what is a root? How are they used with regard to
equations? Different sources seem to be giving me different answers,
some of which contradict each other. What actually IS a root or zero?

I think of a number as a symbol for a measure of magnitude, like how
many of something there are. You can't have 2 + i things, so how can
that be a complex 'number'? How do you define what a number really is?

Why do so many mathematical objects -- groups, fields, rings, vector
spaces -- have nearly identical definitions and properties? Does every
new object have to undergo some rigorous proof of its existence?

Your proof for why x^0 = 1 uses a law which breaks down at x = 0. Then
in your definition for 0^0 you side significantly in favor of 0^0 = 1
based on your rule for x^0 = 1 (which was based on a law that breaks
down at 0). Based on what I've read I would side in favor of
undefined. Are there any more conclusive reasons for siding with 0^0 = 1?